Lorentz transformations and special theory of relativity have existed for more than a century and mathematics related to them has been used and applied for innumerous times. Relativistic energy and relativistic momentum equations have been derived and proven to be conserved if energy/momentum transaction is seen from different frames of reference. The set of permissible inertial reference frame velocities from where the energy and momentum of a closed system of particles may be observed to be conserved forms a ball in the velocity vector space. In this paper we use the existing equations of special theory of relativity and Lorentz transformations and the mathematical structure of the observation velocity space to prove that the conservation of kinetic energy implies the conservation of momentum. We also prove that the conservation of momentum implies the conservation of kinetic energy. We further derive many more linearly independent conservation equations directly from the conservation of energy/ momentum. The derivation of the conservation of kinetic energy from the conservation of momentum implies that either potential energy has a momentum thus made of inertial particles or there cannot be a net conversion of potential energy to kinetic energy. Furthermore the existence of many equations lead to extremely strict form of transfers of energy and momentum. It highly restricts the set of states particles in any closed system can assume without changing the overall energy of the system. This has a strong impact on the particle mechanics and as an example we show that the relativistic explanation of the elastic collision of particles striking each other as used by Einstein in the 1934 two blackboard derivation of mass and energy is itself inconsistent and wrong.